I came across this problem and I had some thoughts. But I don't think I am right.

Suppose $X_i$ are i.i.d random variable with zero mean and unit variance. $S_n = \Sigma_{i=1}^{n} X_i$ Find suitable deterministic sequence $x_n$ and $y_n$ such that $$\frac{((S_n)^2-x_n)}{y_n} $$ convergence in distribution to some non-constant random variable. My idea is that consider $x_n = n$ and $y_n = n$, so we will have the original sequence convergence in distribution to $\frac{S_n-\sqrt{n}}{\sqrt{n}} \frac{S_n-\sqrt{n}}{\sqrt{n}}$ which converges in distribution to $(N(0,1)-1)(N(0,1)+1)$ by using the central limit theorem. But I don't think it is right since we can't naively say that the limit of the product of two random variable is the product of the limit.

Do you guys have any other thoughts? Thanks!


One possibility simply would be to use the continuous mapping theorem. We know

$$ S_n/\sqrt{n} \to X$$

in distribution, where $X \sim N(0,1)$, so

$$g(S_n/\sqrt{n}) \to g(X)$$

in distribution as well, where $g$ is any continuous function. So $g(x) = x^2$ works, and the non-trivial limit is a $\chi_1^2$ distribution.


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